CEE 360/548: Risk Assessment and Management
Prof. Erik VanMarcke
SINGLE-MODE RELIABILITY ANALYSIS
PROBLEM 1
In simple, single-mode reliability analysis, one seeks to evaluate the probability of failure, p_f, defined as the probability that the "load" L exceeds the load-carrying capacity or "resistance" R, where the random variables L and R have the same dimensions and are assumed herein to be statistically independent. In other risk-related applications, L may be the "demand" on a system and R its "capacity"; in the case of a financial institution, L may represent (aggregate) liabilities and R (aggregate) assets.
The probability of failure, p_f = P[L > R], can be expressed in terms of the "margin of safety", Y = R - L,
p_f = P[Y < 0],
or in terms of the "factor of safety", Z = R/L,
p_f = P[Z < 1],
The quantities Y and Z are "derived" random variables that depend on L and R.
Question 1: Express the probability density funtions (pdf's) of Y and Z in terms of the pdf's of L and R. (Hint: see course notes packet, p. 12; first obtain the cumulative probability distribution function, then take its derivative.)
Question 2: Express the pdf of Y in case L and R are both Gaussian (normal), with known means and variances. Also, evaluate the mean and the variance of Y in terms of the means and variances of L and R.
Two useful measures of "the degree of safety" are:
Question 3: Show that p_f has a one-to-one monotonic relationship to "beta" if the safety margin Y has a Gaussian distribution. Sketch the relationship between p_f and "beta", for values of "beta" between 0 and 4. (Hint: use a table of the Standard Normal Cumulative Distribution; see, for instance, p. 20 of the course notes packet).
Question 3-GR (for grad. students registered in CEE 548) : In case L and R are both lognormally distributed, with known means and variances, what is the pdf of Z? Express the mean and the variance of Z. (Hint: see p. 21 in course notes packet for background on the lognormal distribution).
PROBLEM 2 (Continuation)
In the following, assume that the safety margin Y follows a Gaussian distribution and use (the sketch of) the relationship between p_f and "beta" produced in your answer to Problem 1, Question 3.
(a) Risk Assessment
You are given the following information in a "design" situation referred to as "the basic case":
where V denotes the "coefficient of variation", defined as the ratio of the standard deviation to the mean.
Question 4: Evaluate the central safety factor "s", the reliability index "beta", and the probability of failure "p_f".
Question 4-GR (for grad. students registered in CEE 548) : Develop/propose a measure of safety similar to reliability index "beta" for the case when L and R are both lognormally distributed; denote it by "beta prime". (The corresponding central safety measure s' is the difference between the respective logarithms of R and L.) Sketch the relationship between p_f and this (alternate) reliability index for the numerical values of load and resistance statistics as given above.
(b) Risk Management
Often, little can be done about the loads or "demands", but the failure probability can be reduced by changing the probability distribution of the resistance R, either by increasing its mean or by reducing its variability. To illustrate the effects, re-evaluate the quantities s, beta, and p_f for these two cases:
(i) Increase the mean resistance to E[R] = 1.2 m while keeping all other values the same as in "the basic case".
(ii) Through quality control measures, decrease the coefficient of variation of the resistance to V_R = 0.15 while keeping all other values the same as in "the basic case".
Question 5: Which of these two measures most reduces the basic-case failure probability p_f (or increases the basic-case reliability index the most)? Show your calculations.
Note the inadequacy of a common (deterministic) approach in which safety is measured by the "central safety factor" alone, so decisions hinge on whether the outcome is either "s > 1" (safe) or "s < 1" (unsafe).